
Similar models have been used to discuss the coexisting orders and Mott mutilcriticality in Dirac systmes, see Refs. [1]}, [2]}. As defined is Sec. , the \(Z_{i}\)  renormalization factors. We
have also introduced the interaction between the CDW dynamical fluctuating
and QAH dynamical fluctuating, which is given by \(\mathcal {L}_{\chi \phi }\) . Evaluating these renormalization factors at
one-loop order, the beta functions of the rescaled coupling constants are
given by the following differential equations
\(\beta (g_{\chi }^{2})& =-\epsilon g_{\chi }^{2}+(2N_{f}+3)g_{\chi }^{4}+3g_{\phi }^{2}g_{\chi }^{2},  \\\beta (g_{\phi }^{2})& =-\epsilon g_{\phi }^{2}+(2N_{f}+3)g_{\phi }^{4}+3g_{\chi }^{2}g_{\phi }^{2},  \\\beta (\lambda _{\chi })& =-\epsilon \lambda _{\chi }+4N_{f}\lambda _{\chi }g_{\chi }^{2}+36\lambda _{\chi }^{2}-N_{f}g_{\chi }^{4}+\lambda _{\chi \phi }^{2}  \\\beta (\lambda _{\phi })& =-\epsilon \lambda _{\phi }+4N_{f}\lambda _{\phi }g_{\phi }^{2}+36\lambda _{\phi }^{2}-N_{f}g_{\phi }^{4}+\lambda _{\chi \phi }^{2},  \\\beta (\lambda _{\chi \phi })& =-\epsilon \lambda _{\chi \phi }+2N_{f}\lambda _{\chi \phi }g_{\chi }^{2}+2N_{f}\lambda _{\chi \phi }g_{\phi }^{2}+8\lambda _{\chi \phi }^{2} \\& +12\lambda _{\chi }\lambda _{\chi \phi }+12\lambda _{\phi }\lambda _{\chi \phi }-N_{f}g_{\phi }^{2}g_{\chi }^{2}, \) 
